Recall that a finite semigroup S is said to be inherently nonfinitely based (INFB) if S does not belong to any finitely based locally finite variety. In 1987, Mark Sapir proved that the 6-element Brandt monoid B_2^1 is INFB; later he gave an algorithmically efficient description of INFB semigroups. Sapir's description implies, in particular, that no finite J-trivial semigroup is INFB.
The concept of an INFB semigroup has been generalized by Jackson and the speaker as follows: given a class C of semigroup varieties, a finite semigroup S is said to be INFB relative to C if S does not belong to any finitely based variety from C. ("Classic" INFB semigroups are just semigroups which are INFB relative to locally finite varieties.)
In the talk we present a fresh result by Olga Sapir and the speaker that the Catalan monoid C_5, that is, the monoid of all order-preserving and decreasing transformations of the 5-element chain is INFB relative to varieties generated by finite J-trivial semigroups. The proof relies on Simon's celebrated theorem on piecewise testable languages.
We also survey some other interesting equational properties of C_5 which are surprisingly similar to the properties of B_2^1: this part of the talk is based on recent results by Klima, Kunz, and Polak (for C_5) and Jackson (for B_2^1).